Olympiad problems

2011 Today's Calculation Of Integral, 743

Evaluate $\int_0^{\frac{\pi}{2}} \ln (1+\sqrt[3]{\sin \theta})\cos \theta\ d\theta.$

2005 Today's Calculation Of Integral, 34

Tags: probability , integration , calculus , calculus computations

Let $p$ be a constant number such that $0<p<1$. Evaluate \[\sum_{k=0}^{2004} \frac{p^k (1-p)^{2004-k}}{\displaystyle \int_0^1 x^k (1-x)^{2004-k} dx}\]

2010 Today's Calculation Of Integral, 535

Tags: calculus , geometric transformation , geometry , integration , trigonometry , rotation , calculus computations

Let $ C$ be the parameterized curve for a given positive number $ r$ and $ 0\leq t\leq \pi$, $ C: \left\{\begin{array}{ll} x \equal{} 2r(t \minus{} \sin t\cos t) & \quad \\ y \equal{} 2r\sin ^ 2 t & \quad \end{array} \right.$ When the point $ P$ moves on the curve $ C$, (1) Find the magnitude of acceleralation of the point $ P$ at time $ t$. (2) Find the length of the locus by which the point $ P$ sweeps for $ 0\leq t\leq \pi$. (3) Find the volume of the solid by rotation of the region bounded by the curve $ C$ and the $ x$-axis about the $ x$-axis. Edited.

2005 Today's Calculation Of Integral, 58

Tags: calculus computations , calculus , integration

Let $f(x)=\frac{e^x}{e^x+1}$ Prove the following equation. \[\int_a^b f(x)dx+\int_{f(a)}^{f(b)} f^{-1}(x)dx=bf(b)-af(a)\]

Today's calculation of integrals, 855

Tags: calculus computations , limit , function , integration , calculus

Let $f(x)$ be a function which is differentiable twice and $f''(x)>0$ on $[0,\ 1]$. For a positive integer $n$, find $\lim_{n\to\infty} n\left\{\int_0^1 f(x)\ dx-\frac{1}{n}\sum_{k=0}^{n-1} f\left(\frac{k}{n}\right)\right\}.$

2012 Today's Calculation Of Integral, 825

Tags: limit , calculus computations , trigonometry , integration , calculus

Answer the following questions. (1) For $x\geq 0$, show that $x-\frac{x^3}{6}\leq \sin x\leq x.$ (2) For $x\geq 0$, show that $\frac{x^3}{3}-\frac{x^5}{30}\leq \int_0^x t\sin t\ dt\leq \frac{x^3}{3}.$ (3) Find the limit \[\lim_{x\rightarrow 0} \frac{\sin x-x\cos x}{x^3}.\]

2009 Today's Calculation Of Integral, 452

Tags: logarithm , calculus , calculus computations , integration

Let $ a,\ b$ are postive constant numbers. (1) Differentiate $ \ln (x\plus{}\sqrt{x^2\plus{}a})\ (x>0).$ (2) For $ a\equal{}\frac{4b^2}{(e\minus{}e^{\minus{}1})^2}$, evaluate $ \int_0^b \frac{1}{\sqrt{x^2\plus{}a}}\ dx.$

2009 Today's Calculation Of Integral, 438

Tags: calculus computations , trigonometry , integration , calculus

Evaluate $ \int_{\sqrt{2}\minus{}1}^{\sqrt{2}\plus{}1} \frac{x^4\plus{}x^2\plus{}2}{(x^2\plus{}1)^2}\ dx.$

2005 Today's Calculation Of Integral, 20

Tags: calculus , trigonometry , integration , logarithm , calculus computations

Calculate the following indefinite integrals. [1] $\int \ln (x^2-1)dx$ [2] $\int \frac{1}{e^x+1}dx$ [3] $\int (ax^2+bx+c)e^{mx}dx\ (abcm\neq 0)$ [4] $\int \left(\tan x+\frac{1}{\tan x}\right)^2 dx$ [5] $\int \sqrt{1-\sin x}dx$

2011 Today's Calculation Of Integral, 751

Tags: logarithm , integration , trigonometry , calculus , limit , calculus computations

Find $\lim_{n\to\infty}\left(\frac{1}{n}\int_0^n (\sin ^ 2 \pi x)\ln (x+n)dx-\frac 12\ln n\right).$

2008 Teodor Topan, 2

Tags: limit , calculus computations , calculus

Let $ \sigma \in S_n$ and $ \alpha <2$. Evaluate$ \displaystyle\lim_{n\to\infty} \displaystyle\sum_{k\equal{}1}^{n}\frac{\sigma (k)}{k^{\alpha}}$.

2010 Today's Calculation Of Integral, 561

Tags: function , integration , derivative , algorithm , calculus , calculus computations

Evaluate \[ \int_{\minus{}1}^1 \frac{1\plus{}2x^2\plus{}3x^4\plus{}4x^6\plus{}5x^8\plus{}6x^{10}\plus{}7x^{12}}{\sqrt{(1\plus{}x^2)(1\plus{}x^4)(1\plus{}x^6)}}dx.\]

Today's calculation of integrals, 766

Tags: function , inequalities , calculus , integration , trigonometry , calculus computations

Let $f(x)$ be a continuous function defined on $0\leq x\leq \pi$ and satisfies $f(0)=1$ and \[\left\{\int_0^{\pi} (\sin x+\cos x)f(x)dx\right\}^2=\pi \int_0^{\pi}\{f(x)\}^2dx.\] Evaluate $\int_0^{\pi} \{f(x)\}^3dx.$

2007 Today's Calculation Of Integral, 202

Tags: inequalities , integration , trigonometry , calculus computations , calculus , quadratic

Let $a,\ b$ are real numbers such that $a+b=1$. Find the minimum value of the following integral. \[\int_{0}^{\pi}(a\sin x+b\sin 2x)^{2}\ dx \]

2009 Today's Calculation Of Integral, 512

Tags: geometry , integration , calculus , trigonometry , calculus computations

Evaluate $ \int_0^{n\pi} \sqrt{1\minus{}\sin t}\ dt\ (n\equal{}1,\ 2,\ \cdots).$

2010 Today's Calculation Of Integral, 536

Tags: calculus , trigonometry , integration , calculus computations

Evaluate $ \int_0^\frac{\pi}{4} \frac{x\plus{}\sin x}{1\plus{}\cos x}\ dx$.

2007 Today's Calculation Of Integral, 251

Tags: calculus , function , calculus computations , integration , trigonometry

Evaluate $ \int_0^{n\pi} e^x\sin ^ 4 x\ dx\ (n\equal{}1,\ 2,\ \cdots).$

2011 ISI B.Math Entrance Exam, 6

Tags: calculus , trapezoid , integration , geometry , calculus computations , function

Let $f(x)=e^{-x}\ \forall\ x\geq 0$ and let $g$ be a function defined as for every integer $k \ge 0$, a straight line joining $(k,f(k))$ and $(k+1,f(k+1))$ . Find the area between the graphs of $f$ and $g$.

2005 Today's Calculation Of Integral, 8

Tags: function , calculus , trigonometry , integration , logarithm , calculus computations

Calculate the following indefinite integrals. [1] $\int x(x^2+3)^2 dx$ [2] $\int \ln (x+2) dx$ [3] $\int x\cos x dx$ [4] $\int \frac{dx}{(x+2)^2}dx$ [5] $\int \frac{x-1}{x^2-2x+3}dx$

2003 Moldova National Olympiad, 12.1

Tags: limit , calculus , calculus computations , logarithm

For every natural number $n$ let: $a_n=ln(1+2e+4e^4+\dots+2ne^{n^2})$. Find: \[ \displaystyle{\lim_{n \to \infty}\frac{a_n}{n^2}} \].

2006 ISI B.Stat Entrance Exam, 6

Tags: calculus computations , limit , function , calculus

(a) Let $f(x)=x-xe^{-\frac1x}, \ \ x>0$. Show that $f(x)$ is an increasing function on $(0,\infty)$, and $\lim_{x\to\infty} f(x)=1$. (b) Using part (a) or otherwise, draw graphs of $y=x-1, y=x, y=x+1$, and $y=xe^{-\frac{1}{|x|}}$ for $-\infty<x<\infty$ using the same $X$ and $Y$ axes.

2011 Today's Calculation Of Integral, 707

Tags: calculus , calculus computations , integration

In the $xyz$ space, consider a right circular cylinder with radius of base 2, altitude 4 such that \[\left\{ \begin{array}{ll} x^2+y^2\leq 4 &\quad \\ 0\leq z\leq 4 &\quad \end{array} \right.\] Let $V$ be the solid formed by the points $(x,\ y,\ z)$ in the circular cylinder satisfying \[\left\{ \begin{array}{ll} z\leq (x-2)^2 &\quad \\ z\leq y^2 &\quad \end{array} \right.\] Find the volume of the solid $V$.

2010 Today's Calculation Of Integral, 562

Tags: limit , inequalities , integration , calculus , logarithm , calculus computations

(1) Show the following inequality for every natural number $ k$. \[ \frac {1}{2(k \plus{} 1)} < \int_0^1 \frac {1 \minus{} x}{k \plus{} x}dx < \frac {1}{2k}\] (2) Show the following inequality for every natural number $ m,\ n$ such that $ m > n$. \[ \frac {m \minus{} n}{2(m \plus{} 1)(n \plus{} 1)} < \log \frac {m}{n} \minus{} \sum_{k \equal{} n \plus{} 1}^{m} \frac {1}{k} < \frac {m \minus{} n}{2mn}\]

2009 Today's Calculation Of Integral, 457

Tags: calculus , integration , trigonometry , logarithm , calculus computations

Evaluate $ \int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{1}{1\plus{}\sin \theta \minus{}\cos \theta}\ d\theta$

2010 Today's Calculation Of Integral, 626

Tags: limit , calculus , geometry , integration , logarithm , calculus computations

Find $\lim_{a\rightarrow +0} \int_a^1 \frac{x\ln x}{(1+x)^3}dx.$ [i]2010 Nara Medical University entrance exam[/i]